The most famous open question on equidecompositions was Tarski's circle squaring problem [1]}. Laczkovich solved this positively [2]}, but his construction was not measurable, which left the measurable circle squaring problem open for decades.
This was only recently solved by Grabowski, Máthé and Pikhurko [3]}, who found a measurable equidecomposition [3]}, and Marks and Unger [5]} found a Borel equidecomposition.
The measurable circle squaring also can be deduced from the result of the authors [6]} on the existence of measurable perfect matchings in hyperfinite bipartite graphings.
Recently, Máthé, Noel and Pikhurko [7]} managed to find a circle squaring with Jordan measurable pieces which are Boolean combinations of \(F_{\sigma }\)  sets.
